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Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it.
by Ambolia 6y ago
Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it.
- injb 6y ago*complex numbers. But yes you're right. The biggest problem (at least for me) is the name. Because everyone recognizes "real" and "imaginary" as adjectives, they think that's how these words should be understood in the context of numbers. But in truth, they are _proper nouns_, i.e. labels, and nothing more. As far as the adjective "imaginary" is concerned, all numbers are imaginary. Really the sooner we stop calling them "real" and "imaginary" the better. They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either.
- janderson3 6y agoAbsolutely agree. What's your preferred terminology? I like "Forward" and "Reverse" for positive and negative number and "Lateral" for the imaginary unit, since it is just perpendicular to the Forward and Reverse numbers.
- klmadfejno 6y agoGood and evil.
- deleted 6y ago[deleted]
- _jal 6y agoSpooky and strange?
- andrewprock 6y agoHuman and Werewolf.
- kortex 6y agoSpooky Werewolf Algebras on the Monster group. Coming to arxiv this October.
- Robotbeat 6y agoI just think of them as different axes on a 2D Cartesian plane. That’s basically how they work. EDIT: similarly, quaternions as a number in 4D, with that extra dimension being handy for avoiding the instability problems when you get near “gimbal lock” in 3D rotations.
- phkahler 6y agoBut the quaternions sort of make the real component special dont they?
- kortex 6y agoThat it more an effect of how we use them in computer geometry. Unit quaternions with ijk plus real map to an xyz tilt plus a rotor. But I believe you could pick any such mapping. On the other hand, dual quaternions do care which is the dual component, since multiplying by complex numbers looks like true rotation, while multiplying by a dual looks like rotation around a point at infinity, aka translation. So in that case maybe there is something special about the real component.
- injb 6y agoThat's pretty close to what Gauss said they should be called: "If ... +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question" (https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-for-positive-negative-and.html https://shitohichiumaya.blogspot.com/2016/10/gausss-quote-fo...)
- yccs27 6y agoI also like the name "lateral unit", maybe call the field "planar numbers"?
- jonnycomputer 6y agoDo you have a suggestion for an alternative to "real number"?
- gpm 6y agoContinuous numbers?
- macksd 6y agoImaginary numbers are also continuous, are they not?
- gpm 6y agoSo they are, so are many other spaces. I'd avoid calling complex numbers "numbers" personally. In the context of plain english (and not currently in the context of mathematics) I think of the word "number" as saying that this is somehow a measure of a quantity (We have "a number" of brands of cereal in the store). Complex numbers are not a measure of a quantity. I'd probably replace the "numbers" part with "field". I don't have a good replacement for the "complex" part, but I think that should change also (because telling people things are complex is almost as bad as telling them that they are imaginary). If no one proposed something better and I was king of the universe I guess I'd just name them after whoever discovered them.
- injb 6y agoWould you consider -3 a number? It's certainly not a measure of quantity. A better way of thinking about it is that it's 2 things: a magnitude and a direction. Complex numbers just expand the direction into two dimensions. As long as you accept negative numbers, then you already accept the fact that there can be more than one number of a given size. The complex numbers just contain infinitely many numbers of a given size, instead of only 2.
- gpm 6y agoIn plain english, no I don't think I would particularly consider -3 a number. If I say someone gave me a number of horses I don't mean they stole my horse. I think it is the right call to include them in the mathematical definition though. Unlike complex numbers they are almost always useful (they are needed to turn the object into a field, which is needed if you want everyday operations like subtraction to always remain within the set), have a much more intuitive definition of multiplication, are a smaller generalization, etc.
- GoblinSlayer 6y agoIn quantum physics complex numbers are used to represent circular symmetry, so they are more like modulus and phase.
- ko27 6y ago> They are terrible choices of name, coined by people who didn't really understand them, and didn't want you to understand them either. Sorry to be blunt, but your comment is horrible. There is far more going on here than you realize. Real numbers have an isomorphism with our reality. We thought we didn't need imaginary numbers to describe reality. They were inteded to be used more as a mathematical tool. Another good example is euclidean vs non-euclidean geometry. They are both incredibly useful, but the former one was though to be "real" and the latter "imaginary". But it turns, physics is okay with us living in a non-euclidean universe (https://en.wikipedia.org/wiki/Shape_of_the_universe https://en.wikipedia.org/wiki/Shape_of_the_universe).
- fooker 6y ago> Real numbers have an isomorphism with our reality. You can probably get a nobel prize by proving this. :)
- bikenaga 6y agoMathematicians have largely abandoned the term "imaginary number" and use "complex number" instead. I never used the term "imaginary number" when I was teaching (except to point out that it wasn't very good terminology). But the old term still survives in some areas. For instance, the real and imaginary parts of a complex number: re (3 + 4 i) = 3 and im (3 + 4 i) = 4.
- denkquer 6y agoImaginary numbers are 1d. complex are 2d.
- darig 6y agoImagination itself isn't scary if you just think you live in a turtle's dream, flying through space. Granted, my math degree is 20 years old, but I'm pretty sure you're not talking about imaginary numbers. They couldn't exist in 2d.
- rimiform 6y agoThat's great, but my intuition fails me when I try to consider what it would mean for the answer to some question to be a complex number. It's even worse for quaternions, because I can't even use the spatial analogue there.
- munchbunny 6y agoI think about it this way: the complex-ness of the number is really just a representational trick. What it means, intuitively, depends on what the question is really asking and why the behaviors of math on complex numbers is well-suited to representing the answer, and that reason might change based on the context.
- srean 6y agoThere is more to it. Vectors typically don't have multiplication defined. Complex numbers form a field and that is a major extra feature. Multiplication and division I'd defined on them. If this property is not needed one can make do with 2d vectors instead. There are 4d and 8d analogues as well. Hamilton started an ambitious program to base mathematics of nature on the basis of the 4d ones -- quarternions
- gspr 6y ago> Imaginary numbers are not so scary if you just think of them as 2d vectors with a different multiplication operation added to it. Perhaps, but that view makes them seem arbitrary. It makes it seem as if you could just do the same thing in 3d, 4d, etc. and they'd all be equally impacful and meaningful.
- leereeves 6y agoYou can (almost). You can do the same thing again to get quaternions, and again to get octonions, and so on. https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... That's different from extending 1d to 2d, 3d, etc as vectors.
- gspr 6y agoI know. But these are not as fundamentally important as the complex numbers.
- AnimalMuppet 6y ago"And so on"? I thought octonions are where it stops? At least, anything past the octonions is not able to be a normed division algebra, if I understand correctly.
- leereeves 6y agoIt doesn't stop, but as you said, the next step (sedenions), and all steps after that have non-trivial zero divisors and aren't division algebras.
- gspr 6y agoExactly. This is what I was hinting at when critical at people who say complex numbers are "just 2d reals with fancy multiplication". It's true, of course, but it makes it seem as if the 2 in "2d" is unimportant/arbitrary.
- 6y ago
- SavantIdiot 6y agoWhere the analogy to a 2D vector starts to diverge is raising something to a complex power. There's no meaning in raising 3 to the [x,y], but Euler clearly defined it for 3^(xi+y).
- pfortuny 6y agoThe thing (which is subtle) is that the dimensionality is 1. Which is, as far as I have read the intro to the preprint cited in the Quanta article, what the authors are mistaken. Algebraically, as vector spaces: C is the change of base of R when tensored by C which is not the same as R^2 as R-vector space tensored by C (not at all the same).